## Scarred patterns in surface waves

### Arshad Kudrolli, Mathew Abrahams, and Jerry P. Gollub

Phys. Rev. E **63**, 026208 (2001). also: http://arXiv.org/abs/nlin/0002045
High resolution postscript figures: Fig.1. Fig.2. Fig.3. Fig.4. Fig.5. Fig.6. Fig.7. Fig.8. Fig.9. Fig.10.
preprint with figures.

**Abstract:** Surface wave
patterns are investigated experimentally in a system geometry that has become a
paradigm of quantum chaos: the stadium billiard. Linear waves in bounded
geometries for which classical ray trajectories are chaotic are known to give
rise to scarred patterns. Here, we utilize parametrically forced surface waves
(Faraday waves), which become progressively nonlinear beyond the wave
instability threshold, to investigate the subtle interplay between boundaries
and nonlinearity. Only a subset (three main types) of the computed linear modes
of the stadium are observed in a systematic scan. These correspond to modes in
which the wave amplitudes are strongly enhanced along paths corresponding to
certain periodic ray orbits. The suppression of certain other modes has been
shown by Agam and Altshuler in a related theoretical study to be a consequence
of higher boundary dissipation relative to the modes that are observed.
Spatially asymmetric or disordered (but time-independent) patterns are also
found even near onset. As the driving acceleration is increased, the
time-independent scarred patterns persist, but in some cases transitions between
modes are noted. The onset of spatiotemporal chaos at higher forcing amplitude
often involves a nonperiodic oscillation between spatially ordered and
disordered states. We have characterized this phenomenon using the concept of
pattern entropy. The rate of change of the patterns is found to be reduced as
the state passes temporarily near the ordered configurations of lower entropy.
We also have found complex but highly symmetric (time-independent) patterns far
above onset in the regime that is normally chaotic.

**Movies of time-dependent surface wave patterns**

Aperiodic oscillation between spatially ordered and disordered patterns is observed (driving frequency = 55 Hz).

*Interesting Quantum Chaos related links: *

Simulation of ray trajectories in a stadium (and circle.) Link.

Please send comments to:

Arshad Kudrolli

akudrolli@clarku.edu

and Jerry
Gollub

jgollub@haverford.edu

Last modified Feb. 24, 2000.